Some remarks on Quillen's Dévissage theorem

TL;DR

The paper proves a new, conceptual route to Quillen's dévissage by deriving it from Barwick's theorem of the heart, via a short exact sequence of dg categories built from a quasi-abelian pair ${\mathcal{A}}\supset{\mathcal{B}}$ and the auxiliary category ${\mathcal{E}}_{\mathcal{A},\mathcal{B}}$. It introduces the right abelian envelope ${\mathcal{L}}{\mathcal{H}}({\mathcal{E}}_{\mathcal{A},\mathcal{B}})$ through an explicit abelian model ${\mathcal{C}}_{\mathcal{A},\mathcal{B}}$, identifies a torsion pair with ${\mathcal{B}}$ as a torsion subcategory, and shows a Serre quotient recovers ${\mathcal{A}}$. Combining these constructions with Barwick’s $K$-theory results yields $K_n({\mathcal{B}})\xrightarrow{\sim} K_n({\mathcal{A}})$ for all $n\ge 0$, with a concrete description of $K_0$-levels via a universal inverse. The approach also connects to Auslander-type constructions for nilpotent extensions and aligns with prior work of Kuznetsov–Lunts, Land–Tamme, and the author, offering a streamlined, category-theoretic pathway to dévissage.

Abstract

In this paper we give a different proof of Quillen's Dévissage theorem using Barwick's theorem of the heart. The key ingredient is a certain short exact sequence of dg categories, which is closely related with the Auslander-type construction for nilpotent extensions which was used in the papers of Kuznetsov-Lunts \cite{KL15}, Land-Tamme \cite{LT19} and the author \cite{E20}.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4
• proof
• Corollary 1.5
• proof